The analytic hierarchy process, max algebra and multi-objective optimisation

The analytic hierarchy process (AHP) is widely used for decision making involving multiple criteria. Elsner and van den Driessche (2004, 2010) [10,11] introduced a max-algebraic approach to the single criterion AHP. We extend this to the multi-criteria AHP, by considering multi-objective generalisations of the single objective optimisation problem solved in these earlier papers. We relate the existence of globally optimal solutions to the commutativity properties of the associated matrices; we relate min–max optimal solutions to the generalised spectral radius; and we prove that Pareto optimal solutions are guaranteed to exist.